Ch 22: Gauss' Law

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Non è quello che usi tu?Cambia libro di testo

Young & Freedman Calc 15th Edition

Ch 22: Gauss' Law

Problema 24b

Capitolo 22, Problema 24b

Charge $q$ is distributed uniformly throughout the volume of an insulating sphere of radius $R = 4.00$ cm. At a distance of $r = 8.00$ cm from the center of the sphere, the electric field due to the charge distribution has magnitude $E = 940$ N/C. What is the electric field at a distance of $2.00$ cm from the sphere's center?

Guida verificata passo dopo passo

Understand that the problem involves a uniformly charged insulating sphere, which means we can use Gauss's Law to find the electric field at different distances from the center.

Recall Gauss's Law: $ \Phi_E = \frac{Q_{enc}}{\varepsilon_0} $, where $ \Phi_E $ is the electric flux, $ Q_{enc} $ is the enclosed charge, and $ \varepsilon_0 $ is the permittivity of free space.

For a point inside the sphere (at 2.00 cm from the center), the enclosed charge $ Q_{enc} $ is proportional to the volume of the sphere up to that radius. Use the formula $ Q_{enc} = \frac{q}{V_{total}} \times V_{enc} $, where $ V_{total} $ is the total volume of the sphere and $ V_{enc} $ is the volume enclosed by the radius 2.00 cm.

Calculate the volume of the sphere using $ V = \frac{4}{3} \pi R^3 $ and the volume enclosed by the radius 2.00 cm using $ V_{enc} = \frac{4}{3} \pi r^3 $.

Apply Gauss's Law to find the electric field $ E $ at 2.00 cm: $ E = \frac{Q_{enc}}{4\pi \varepsilon_0 r^2} $, where $ r $ is the distance from the center (2.00 cm in this case).

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Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is expressed as Φ = Q_enclosed/ε₀, where Φ is the electric flux, Q_enclosed is the charge within the surface, and ε₀ is the permittivity of free space. This law is crucial for calculating electric fields in symmetric charge distributions, such as spheres.

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Gauss' Law

Electric Field Inside a Uniformly Charged Sphere

For a uniformly charged insulating sphere, the electric field inside the sphere (at a distance r from the center) is given by E = (1/4πε₀) * (Q_enclosed/r²), where Q_enclosed is the charge within a sphere of radius r. This field increases linearly with distance from the center until reaching the sphere's surface.

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Electric Field due to a Point Charge

Charge Density and Enclosed Charge

Charge density (ρ) is the charge per unit volume, given by ρ = Q/V for a sphere. To find the charge enclosed within a smaller sphere of radius r, use Q_enclosed = ρ * (4/3)πr³. This concept helps determine the electric field at any point inside the sphere by calculating the charge enclosed within that radius.

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Charging Objects

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